If sync is dropping when you go into a menu and then comes back a few seconds later, it’s likely because the resolution is switching from 240p (gameplay) to 480i (menus) and back, and sync is briefly lost. On a CRT, you don’t notice the resolution change; but, when sync is lost over HDMI, all the devices between your OSSC and display need to do another handshake before anything is shown on the screen, which can take a few seconds.
There’s not too much you can do besides perhaps configuring your OSSC for something like 2x for both 240p and 480i (bob deinterlace), then run your OSSC into an EDID minder set for 480p, and even that might not be perfect.